Proposition 4 (Th. 4, [1]})  With \(\beta _0 = \frac{2 D_{\mathcal {P}} {b}_1}{\Vert {w}\Vert } \)  and \(\beta _t = \frac{\beta _0}{\sqrt{t}}\) , FW-smoothing obtains the following convergence rate:
\(f(P^{(T)}) - f(P^*) \le \frac{2 D_{\mathcal {P}} {b}_1 \Vert {w}\Vert }{\sqrt{T}}.\) 

